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Homework Help: Factor a 3rd order polynomial

  1. Jul 4, 2012 #1
    1. The problem statement, all variables and given/known data
    What are the steps to factoring 3rd order polynomials like x^3+8x^2-21x+10?
    It's to find eigenvalues of a matrix in linear algebra, I completely forgot how to factor and it's killing me.

    2. Relevant equations

    3. The attempt at a solution
    None, unless its a polynomial that I can factor by grouping, I have no clue how to begin.
  2. jcsd
  3. Jul 4, 2012 #2


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    EDIT: Nevermind the below, that factor trick (the 'rational root test') doesn't work in this case. The discriminant is positive, but it turns out that all three roots are irrational.

    For a cubic polynomial with integer coefficients, which is what you have, I believe one of the roots is typically a factor of the constant term. So, try the factors of 10 and see if one of them is a root. Once you've figured out one of the roots, you can factor an (x-root) term out by polynomial division, leaving you with (x-root)*quadratic, and the quadratic you can factor with the quadratic formula.
    Last edited: Jul 4, 2012
  4. Jul 4, 2012 #3
    :( so the pq thing with synthetic and what-the-other-one's-name division?
    I dont know why i didnt think of it (but that sounds really tedious).
    Thanks for your help! :)
  5. Jul 4, 2012 #4
    Got it, so i happened to pick the only example in the book of a matrix with irrational eigenvalues. I'm still not so sure how to find the irrational roots of a 3rd order polynomial... i should change the name of this thread haha.
  6. Jul 4, 2012 #5


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    There's always the cubic formula, but using it is a bit... tedious. :eek:
  7. Jul 7, 2012 #6
    To add to what "eumyang"" said, using the Cubic Formula to get an exact algebraic result is, indeed, a bit tedious.

    On the other hand, if you are fine with numerical results, there are several free online utilities available; some of them are listed at the bottom of that Wikipedia page.

    In addition, I refer you to an old post I made at the bottom of the following thread:


    A search of these forums would probably turn up much more information about factoring a cubic polynomial.
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